Tutorial: Introduction to Dimensional Analysis

The assumption is that you've gone over scientific notation, significant figures,
and the metric system before you begin this tutorial.

Mention "dimensional analysis", "factor-label method", or "unit analysis" to
students and you can sometimes hear the cries of dismay. I think this area is
often terribly misunderstood by many students. It's unfortunate that many
people only have their eye on the end-result rather than how they got to that
point. As a teaching assistant at The Ohio State University, I'll never forget
one situation with a student. The university has a room where all the general
chemistry TA's hold office hours which allows students to have access to people
other than their own TA for help. One day, a student walked up and asked how
to get the answer to a lab report question. As I proceeded to work with her to help
her understand how to work the question, she stopped me and said- "this is due in
15 minutes and I only want to know the answer." Well, I stopped and told
her she should ask someone else because I didn't work that way. And she
then walked away in a huff and probably found someone else. A case of not
seeing the forest for the trees, I suppose...

Dimensional analysis is one of the most useful tools that students will learn in
a science class. Not only does it encourage critical thinking, but it's something
that's used every day by everybody. Yes, I mean everybody. You've already used
this concept even though you may not have realized it. That quick calculation when
you pulled up to the gas station with $5 in your pocket and you estimated how
many gallons of gas you could get. Which laundry detergent is the best bargain? Just
how many US dollars are in that Canadian dollar you're about to use in Quebec? This
is not just a topic for scientific calculations.

An excellent example of just how dimensional analysis impacts our lives came
in 1999 with one of NASA's Mars missions. After the loss of the Climate Observer,
an investigation board was put together to determine what happened to the
mission and a number of things were found to be responsible for inadequate
checks and such. But, there was one factor that directly resulted in the mission's
failure. I'll quote from the press release so you can read it in NASA's own words:

"The 'root' cause of the loss of the spacecraft was the failed translation
of English units in a segment of ground-based, navigation-related mission software,
as NASA has previously announced," said Arthus Stephenson, chairman of the Mars
Climate Orbiter Mission Failure Investigation Board.

The problem was that one team used English units (feet, pounds, etc.) while
another team used metric units which caused the craft to take a different
path than intended. In other words, there was a problem in unit conversions- what
you're learning right now!

Improper dimensional analysis led not only to a massive loss of scientific
knowledge to the world but millions of dollars of United State's citizens' tax
money. See, dimensional analysis really does affect everyone!